A motion graph is just a picture of a journey. Instead of a table of numbers, you get a line whose shape tells you everything — when the object sped up, slowed down, stopped, or turned around. Once you learn to read two simple things — the gradient (steepness) and the area underneath — these graphs hand you velocity and displacement for free. Let’s learn to read them like a story.
📘 What you need to know
There are three motion graphs: displacement–time, velocity–time, and acceleration–time
On a displacement–time graph, the gradient = velocity
On a velocity–time graph, the gradient = acceleration, and the area under = displacement
On an acceleration–time graph, the area under = change in velocity
A straight line means a steady rate; a curved line means the rate is changing
Find a gradient with a large triangle; find an area by splitting it into rectangles and triangles
The two things that unlock every graph
Before we look at each graph, hold on to two ideas. They do all the heavy lifting.
gradient (steepness)
→ tells you →
the rate of change
area underneath
→ tells you →
the total built up
The gradient answers “how fast is the y-axis quantity changing?” The area answers “how much of the y-axis quantity has piled up over time?” Which one is useful depends on which graph you’re looking at — so let’s take them one at a time.
Displacement–time graphs
Here displacement is on the y-axis, time on the x-axis. The key fact: the gradient is the velocity (because velocity is displacement ÷ time, which is exactly rise ÷ run).
A straight, sloping line → constant velocity.
A curved line → the velocity is changing, so the object is accelerating.
A steeper line → a faster velocity; a horizontal line → at rest (not moving).
A negative slope → moving back in the negative direction. The area underneath means nothing here.
Velocity–time graphs
Now velocity is on the y-axis. This is the most useful graph of the three, because it gives you two things: the gradient is the acceleration, and the area underneath is the displacement.
A straight, sloping line → uniform (constant) acceleration.
A curved line → non-uniform acceleration.
A horizontal line → constant velocity (zero acceleration).
The area between the line and the time axis is the displacement travelled.
On a velocity–time graph
gradient = acceleration • area underneath = displacement
Acceleration–time graphs
Least common of the three. Acceleration sits on the y-axis. Here the gradient isn’t useful, but the area underneath equals the change in velocity. A flat horizontal line simply means the acceleration is constant.
How the three graphs connect
These three graphs are really three views of the same journey. The gradient of one becomes the y-axis of the next. Read across the row below: a constant velocity, a steady acceleration, and a growing acceleration, each shown in all three graph types.
The same constant-velocity journey seen three ways — taking the gradient of each graph gives you the next one along.
Here’s the mental shortcut I give students: going right, you take gradients; going left, you find areas. Displacement → velocity → acceleration by gradient each step. And backwards: acceleration → velocity → displacement by area each step. If you can hold that one sentence, you’ll never mix up which tool a graph needs.
Reading a gradient off a graph
To get a velocity from a displacement–time graph, or an acceleration from a velocity–time graph, you find the gradient — rise divided by run. The golden rule: draw a big triangle. A large triangle keeps your reading errors small.
Acceleration is the gradient of a velocity–time line: rise (12 m s−1) ÷ run (6 s) = 2 m s−2.
WE 1
On the velocity–time graph above, the line rises from 3 m s−1 at t = 4 s to 15 m s−1 at t = 10 s. Find the acceleration.
Acceleration = gradient = rise ÷ run
rise = 15 − 3 = 12 m s⁻¹
run = 10 − 4 = 6 s
gradient = 12 ÷ 6acceleration = 2 m s⁻²A big triangle (6 s wide) keeps the reading accurate.
Reading an area off a graph
To get a displacement from a velocity–time graph, find the area between the line and the time axis. If the shape is awkward, chop it into rectangles and triangles, work out each piece, and add them up.
📐 Areas of the shapes you’ll need
Rectangle (constant velocity) → area = base × height.
Triangle (starting or ending at rest) → area = ½ × base × height.
Trapezium (a mix) → split it into a rectangle plus a triangle and add.
WE 2
A car starts from rest and accelerates uniformly to 24 m s−1 over 12 s. Using the velocity–time graph, find the displacement.
Displacement = area under the line
the graph is a triangle (starts at rest)
Area of a triangle = ½ × base × height
base = 12 s, height = 24 m s⁻¹
= ½ × 12 × 24displacement = 144 m
WE 3
A runner accelerates from rest to 20 m s−1 in 8 s, then holds 20 m s−1 for a further 6 s. Find the total displacement.
Split the area into two shapesTriangle (the speeding-up part)
= ½ × 8 × 20 = 80 m
Rectangle (the steady part)
= 20 × 6 = 120 m
Add them together80 + 120total displacement = 200 m
💡 Top tips
Check the axes first — a straight line means very different things on a displacement graph versus a velocity graph.
Use a large gradient triangle — the bigger the rise and run, the smaller your error.
Split awkward areas into rectangles and triangles, then add — don’t try to eyeball the whole shape.
Watch for negative areas — if a velocity line dips below the time axis, that displacement counts as negative.
⚠ Common mistakes
Mixing up the graphs — taking the area of a displacement–time graph (it means nothing there)
Reading a gradient off a tiny triangle and magnifying the error
Thinking a horizontal line always means “stopped” — on a velocity graph it means constant velocity
Forgetting that a line below the axis gives negative velocity or negative displacement
Up next: Projectile Motion — objects moving through the air under gravity, where you split the motion into independent horizontal and vertical parts and apply the SUVAT equations to each.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
